## Differential ManifoldsThe present volume supersedes my Introduction to Differentiable Manifolds written a few years back. I have expanded the book considerably, including things like the Lie derivative, and especially the basic integration theory of differential forms, with Stokes' theorem and its various special formulations in different contexts. The foreword which I wrote in the earlier book is still quite valid and needs only slight extension here. Between advanced calculus and the three great differential theories (differential topology, differential geometry, ordinary differential equations), there lies a no-man's-land for which there exists no systematic exposition in the literature. It is the purpose of this book to fill the gap. The three differential theories are by no means independent of each other, but proceed according to their own flavor. In differential topology, one studies for instance homotopy classes of maps and the possibility of finding suitable differentiable maps in them (immersions, embeddings, isomorphisms, etc.). One may also use differentiable structures on topological manifolds to determine the topological structure of the manifold (e.g. it la Smale [26]). |

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### Contents

Chapter I | 1 |

Integration and Taylors formula | 10 |

Chapter II | 21 |

Copyright | |

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### Common terms and phrases

apply assume ball Banach space boundary bounded called Chapter chart class Cp closed compact conclude consider constant contained coordinates Corollary covering define definition denote derivative determined differential equation differential form domain element equal exists expression exterior derivative fact fiber fixed flow follows formula function Furthermore give given Hence identity induces initial condition integral curve interval inverse isomorphism Lemma lies locally manifold mean value theorem means measure morphism obtain obvious once open neighborhood open set open subset operators oriented partitions of unity positive projection Proof properties Proposition prove representation respectively satisfies sequence shows splits structure submanifold Suppose symmetric tangent bundle theorem topological trivialisation unique variables VB-morphism vector bundle vector field vector space write